The Verdict: Not eligible
An engagement or relationship gives no route to Turkish citizenship; the three-year clock starts only once you are married.
Article 16 of the Turkish Citizenship Law (Law 5901) lets a foreigner who has been married to a Turkish citizen for at least three years, and is still married, apply for citizenship. The conditions are that you live together as a family, do nothing incompatible with the marriage, and pose no national security or public order obstacle. It applies to husbands and wives alike, and the law does not require you to live in Turkey.
Once you marry, register the marriage in the Turkish civil register (through a Turkish consulate if you marry abroad). Until then, our other Turkey quizzes cover residence through property, investment or work.